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Product Rule


The derivative identity

d/(dx)[f(x)g(x)]=lim_(h->0)(f(x+h)g(x+h)-f(x)g(x))/h
(1)
=lim_(h->0)[(f(x+h)g(x+h)-f(x+h)g(x))/h+(f(x+h)g(x)-f(x)g(x))/h]
(2)
=lim_(h->0)[f(x+h)(g(x+h)-g(x))/h+g(x)(f(x+h)-f(x))/h]
(3)
=f(x)g^'(x)+g(x)f^'(x),
(4)

where f^'(x)=df/dx denotes the derivative of f. The Leibniz identity extends the product rule to higher-order derivatives.


See also

Chain Rule, Derivative, Exponent Laws, Leibniz Identity, Quotient Rule, Related Rates Problem

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References

Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 11, 1972.

Cite this as:

Weisstein, Eric W. "Product Rule." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ProductRule.html

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